DeSmuMe supports it. If you're on Windows, go to Config -> Slot 2 (GBA Slot) and then you can pick the GBA ROM you want to have in slot 2. Then launch HG/SS and you should be able to use the Pal Park or whatever it's called (I forgot). After that, to go to Gen 5/6, you'd need to use the function in Black/White (2) to move them to Gen 5, then Bank or whatever to move them to Gen 6. You might need to use a save editor to go from 4 -> 5 though, on an emulator; I don't think DeSmuMe does local multiplayer with 2 instances of the emulator, but I could be wrong.

If you want to move Gen 1 or Gen 2 to Gen 3 and beyond, that's quite a bit more complicated, needing save editors and stuff to get the raw Pokémon's data.

 

I'm not really sure what sort of converters are out there to be honest. Does OSX even come with Java installed anymore? That might be why OSX told you it was "damaged" (Java might be missing or something). And yes, Desmume has an OSX port, but I don't know if it has all of the options the Windows build does.

 

Or you can just use pkhex on pc, load up a .pkm file, open it and set it to one of your spots in a box on omega or sapphire, then export your save file, import the save back into your game, load it up and enjoy your older gen pokémon on your gen VI game.

 

Short answer: As long as youre working with generation 3, 4 or 5, its possible. (Gba and nds games)
Generations 1 and 2 wont work. (Gb and gbc) neither will gen 6 (3ds)

Long answer: See daxtsu's answer above about using your emulator to link games. You can move gen 3 to gen 4 easily. The following method may also work to move from gen 3 to gen 4, but im pretty sure gen3 pkm files arent directly compatable with higher generations.

Once the pokemon is in gen 4, it should be easy to extract the pkm file from the gen 4 sav file using pokesav. And just import it into the sav of the gen 5 game using pokesav. (There are different pokesav programes for each game).

You will not be able to move pokemon from gen 5 to gen 6 using an emulator. This is because pokemon are stored differently from before and there is no easy way to transfer them over with that method. However, if you have access to a physical gen 5 game, you can inject your emulator sav file into the physical cart using a variety of methods. And then transfer them legitimately to gen 6.
See the answer above about pkhex. I dont know much about it, but if its an exe, you can use wine or vmware to run it on osx.

I will also mention that it will be completely impossible to move pokemon from gen1 or 2 to any higher generation. This is because pokemon data is stored in a different manner to make up for abilities, and the special stat split and a whole host of other things (thats the short answer) you could attempt to recreate the pokemon using a pokesav for the generation you want, but you will have to make up/guestimate iv and ev values.

Im on mobile and so I cant give you all the links you might need, but they will be super easy to find if you use google.

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Almost all of these programs are windows programs btw, definately look into getting wine, or windows xp running in a virtual machine on your mac. I use winebottler for most of my exe files. There are some that just dont work, so I just use an old junk laptop I have for those

 

Thank-you for all the input. It is going to help out a lot!
It seems like I am pretty much good to go from here on out with this subject.
As well as all other pkmn trainers out there that want to keep their teams and trade them over.

-To be honest, I have only recently begun replaying pokemon (using retroarch on my n3ds). I found out the hard way after defeating the 3rd gym member in Pokemon Crystal (Gen II) for Gameboy (on Gambatte) that there is no "REAL" way to transfer your legit pokemon to Generation III (due to the IV and EV stats changing, which I do not understand yet). So I am going to start with LeafGreen (Gpsp) and when I am done I would like to trade my LeafGreen Pokemon to HeartGold or SoulSilver (once I decide on a DS Flashcard) (But I will obviously use my Mac and Emulators to conduct the trade and import the saves) and start playing the remakes of Gen II (HGSS) and so on.. and so on, Gen II trading to Gen III Emerald and so on.

One last question about save files. Not every emulator uses the same kind of save files, I notice some of them are different depending on the emulator. Will that be a problem if I try to migrate a game save with a different extension to another emulator? For example since I am using RetroArch and once I migrate the saves to my computer?